3 Answers
3

In your Illustrator document first type the infinity symbol with the Type tool.

Typing infinity symbol: easy way

Type an 8.

Rotate it 90 degrees.

Typing infinity symbol: elaborate way

Go to Type -> Glyphs.

Scroll through all the symbols to find the infinity symbols

Double click it.

Creating path from text

Type the infinity symbol using the above method. I used the second method and typed the infinity symbol from the Arial font. On a mac there is a keyboard shortcut for the symbol (Option+5), don't know if Windows has an equivalent.
I arbitrarily chose a font size of 350pt

Select the text and make an outline of it (Type ->Create Outline`)

Keeping the outlines selected make the fill transparent and give the stroke a colour.

In the Layers panel unfold the layer and the group and select the Compound Path by clicking on the circle next to it.

Go to Object -> Compound Path -> Release

Delete the two inner circles.

Select the remaining outer path.

Go to Object -> Path -> Offset Path and fill in the values shown below.
Keep in mind that this is valid for the Arial infinity symbol at a size of 350pt. Another font and size might need other setting, you can play around with it.

In the Layers panel drag your newly created offset shape out of the group and delete the remaining path. Now you have the infinity symbol path.

Note: If you use the simple method of typing the infinity symbol you will need to manually adjust the path after step 8 such that the middle point becomes one. This is very easily done.

I managed it by creating two circles, cutting them at facing points, then manually joining the cut ends to each other in an overlapping fashion. The downside is that with this method you need to re-adjust the handles on the top and bottom anchor to restore a circular shape to each loop.